If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about the equilibrium point as it does so. In contrast, a critically damped system moves as quickly as possible towards equilibrium without oscillating about the equilibrium point.
Generally, critical damping is often desired because such a system not only returns to equilibrium rapidly, but remains at equilibrium too. In addition, a constant force applied to a critically damped system moves the system to a new equilibrium position in the shortest time possible, without overshooting or oscillating about the new position. For example, when a person stands on a bathroom scale that has a needle gauge, the needle moves to its equilibrium position without oscillating. It would be quite inconvenient if the needle oscillated about the new equilibrium position for a long time before settling. Damping forces can vary greatly in character. Friction, for example, is sometimes independent of velocity. However, many damping forces depend on velocity—sometimes in complex ways and sometimes simply being proportional to velocity.
This text is adapted from Openstax, College Physics, Section 16.7: Damped Harmonic Motion and Openstax, University Physics Volume 1, Section 15.4: Damped Oscillations.
